Theorem exgenall-P6 732

Description: Dual of gennall-P6 730.

See exgenallw-P6 680 for a version that requires only FOL axioms.

Assertion

Ref	Expression
exgenall-P6	⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem exgenall-P6

Step	Hyp	Ref	Expression
1		gennall-P6 730	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
2		lemma-L5.01a 600	. 2 ⊢ ((∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑))
3	1, 2	bimpr-P4.RC 534	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-L10 27

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596

This theorem is referenced by:  gennfr-P6  734  gennfrcl-L6  812